For equilibrium of the system, value of mass $m$ should be .......... $kg$

Find minimum height of obstacle so that the sphere can stay in equilibrium. 

A non uniform cylinder of mass $m$ , length $l$ and radius $r$ is having its cetnre of mass at a distance $l/4$ from the centre and lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density $\rho $ . The moment of inertia of the rod about the centre of mass is $I$ . The angular acceleration of point $A$ relative to point $B$ just after the rod is released from the position shown in figure is

A uniform meter scale is supported from its $20\ cm$ mark. A body suspended from $10\ cm$ mark keeps the scale horizontal. However, the scale gets unbalanced if the body is completely immersed in water. To regain the balance the body is shifted to the $8\ cm$ mark. Therefore, the specific gravity of the material of the body is

A mass $m$ hangs with the help of a string wrapped around a pulley on a firctionless  bearing. The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect  uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the  pulley, is:-

$ABC$ is an equilateral triangle with $O$ as its centre. $\vec F_1, \vec F_2 $and $\vec F_3$ represent three forces acting along the sides $AB, BC$ and $AC$ respectively. If the total torque about $O$ is zero then the magnitude of  $\vec F_3$ is